Three-dimensional (3D) bioprinting is an appealing and revolutionary manufacturing approach for the accurate placement of biologics, such as living cells and extracellular matrix (ECM) components, in the form of a 3D hierarchical structure to fabricate synthetic multicellular tissues. Many synthetic and natural polymers are applied as cell printing bioinks. One of them, alginate (Alg), is an inexpensive biomaterial that is among the most examined hydrogel materials intended for vascular, cartilage, and bone tissue printing. It has also been studied pertaining to the liver, kidney, and skin, due to its excellent cell response and flexible gelation preparation through divalent ions including calcium. Nevertheless, Alg hydrogels possess certain negative aspects, including weak mechanical characteristics, poor printability, poor structural stability, and poor cell attachment, which may restrict its usage along with the 3D printing approach to prepare artificial tissue. In this review paper, we prepare the accessible materials to be able to encourage and boost new Alg-based bioink formulations with superior characteristics for upcoming purposes in drug delivery systems. Moreover, the major outcomes are discussed, and the outstanding concerns regarding this area and the scope for upcoming examination are outlined.